3.8.86 \(\int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx\) [786]

Optimal. Leaf size=35 \[ -\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}} \]

[Out]

-1/3*a/x^2/(c*x^2)^(1/2)-1/2*b/x/(c*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \begin {gather*} -\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^3*Sqrt[c*x^2]),x]

[Out]

-1/3*a/(x^2*Sqrt[c*x^2]) - b/(2*x*Sqrt[c*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x^4} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^4}+\frac {b}{x^3}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 22, normalized size = 0.63 \begin {gather*} \frac {c (-2 a-3 b x)}{6 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x^3*Sqrt[c*x^2]),x]

[Out]

(c*(-2*a - 3*b*x))/(6*(c*x^2)^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 21, normalized size = 0.60

method	result	size
risch	\(\frac {-\frac {b x}{2}-\frac {a}{3}}{x^{2} \sqrt {c \,x^{2}}}\)	\(20\)
gosper	\(-\frac {3 b x +2 a}{6 x^{2} \sqrt {c \,x^{2}}}\)	\(21\)
default	\(-\frac {3 b x +2 a}{6 x^{2} \sqrt {c \,x^{2}}}\)	\(21\)
trager	\(\frac {\left (-1+x \right ) \left (2 a \,x^{2}+3 x^{2} b +2 a x +3 b x +2 a \right ) \sqrt {c \,x^{2}}}{6 c \,x^{4}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^3/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(3*b*x+2*a)/x^2/(c*x^2)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 19, normalized size = 0.54 \begin {gather*} -\frac {b}{2 \, \sqrt {c} x^{2}} - \frac {a}{3 \, \sqrt {c} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*b/(sqrt(c)*x^2) - 1/3*a/(sqrt(c)*x^3)

________________________________________________________________________________________

Fricas [A]
time = 1.14, size = 23, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (3 \, b x + 2 \, a\right )}}{6 \, c x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

-1/6*sqrt(c*x^2)*(3*b*x + 2*a)/(c*x^4)

________________________________________________________________________________________

Sympy [A]
time = 0.22, size = 29, normalized size = 0.83 \begin {gather*} - \frac {a}{3 x^{2} \sqrt {c x^{2}}} - \frac {b}{2 x \sqrt {c x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**3/(c*x**2)**(1/2),x)

[Out]

-a/(3*x**2*sqrt(c*x**2)) - b/(2*x*sqrt(c*x**2))

________________________________________________________________________________________

Giac [A]
time = 2.69, size = 20, normalized size = 0.57 \begin {gather*} -\frac {3 \, b x + 2 \, a}{6 \, \sqrt {c} x^{3} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/6*(3*b*x + 2*a)/(sqrt(c)*x^3*sgn(x))

________________________________________________________________________________________

Mupad [B]
time = 0.15, size = 26, normalized size = 0.74 \begin {gather*} -\frac {2\,a\,\sqrt {x^2}+3\,b\,x\,\sqrt {x^2}}{6\,\sqrt {c}\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(x^3*(c*x^2)^(1/2)),x)

[Out]

-(2*a*(x^2)^(1/2) + 3*b*x*(x^2)^(1/2))/(6*c^(1/2)*x^4)

________________________________________________________________________________________